Dispersal can have major consequences for individual fitness, population dynamics, and species’ distributions. In light of the important need to predict how populations will respond to invasion and spread of alien species, habitat fragmentation, and climate change, understanding the causes and consequences of dispersal at both the patch and landscape levels is vital for population management and conservation.
As the impact of dispersal on population dynamics is becoming increasingly recognized, the consequences and causes of dispersal have become a focus of much current research. Theoretical studies play an extremely important role in predicting the population level effects of dispersal. However, the assumptions of most theoretical models regarding the dispersal process often lack a great deal of realism. The paradigmatic view in ecology is that emigration is density independent (DIE) or positive density dependent (+DDE).
However, alternative forms are biologically plausible, including negative (-DDE), u-shaped (uDDE), and hump-shaped (hDDE) forms (Fig 1).
With funding from the NSF Mathematical Biology Program in 2015 and a renewal of our grant in 2019, we have been examining how different forms of DDE, habitat fragmentation, competition (intra- and interspecific) and predation can affect the population dynamics of natural and model systems. Experimental studies are being led by PI James T. Cronin and his lab (Department of Biology, Louisiana State University), using Tribolium flour beetles as a model system. Mathematical models are being developed by PI Ratnasingham Shivaji (Department of Mathematics & Statistics, The University of North Carolina at Greensboro) and me (see photos below).
Even though mutualistic interactions are ubiquitous in nature, we are still far from making good predictions about the fate of mutualistic communities under threats such as habitat fragmentation and climate change. Fragmentation often causes declines in abundance of a species due to increased susceptibility to edge effects between remnant habitat patches and lower quality “matrix” surrounding these focal patches. It has been argued that ecological communities are replete with trait-mediated indirect effects, and that these effects may sometimes contribute more to the dynamics of a population than direct density-mediated effects, e.g., lowering an organism’s fitness through competitive interactions. Although some studies have focused on trait-mediated behavior such as trait-mediated dispersal, in which an organism changes its dispersal patterns due to the presence of another species, they have been mostly limited to predator-prey systems–little is known regarding their effect on other interaction systems such as mutualism. Here, we explore consequences of fragmentation and trait-mediated dispersal on coexistence of a system of two mutualists by employing a model built upon the reaction diffusion framework. To distinguish between trait-mediated dispersal and density-mediated effects, we isolate effects of trait-mediated dispersal on the mutualistic system by excluding any direct density-mediated effects in the model. Our results demonstrate that fragmentation and trait-mediated dispersal can have important impacts on coexistence of mutualists. Specifically, one species can be better able to invade and persist than the other and be crucial to the success of the other species in the patch. Matrix quality degradation can also bring about a complete reversal of the role of which species is supporting the other’s persistence in the patch, even as the patch size remains constant. As most mutualistic relationships are identified based on density-mediated effects, such an effect may be easily overlooked.
Emigration is a fundamental process affecting species’ local, regional, and large-scale dynamics. The paradigmatic view in ecology is that emigration is density independent (DIE) or positive density dependent (+DDE). However, alternative forms are biologically plausible, including negative (−DDE), U-shaped (uDDE), and hump-shaped (hDDE) forms. We reviewed the empirical literature to assess the frequency of different forms of density-dependent emigration and whether the form depended on methodology. We also developed a reaction-diffusion model to illustrate how different forms of DDE can affect patch-level population persistence. We found 145 studies, the majority representing DIE (30%) and +DDE (36%). However, we also regularly found −DDE (25%) and evidence for nonlinear DDE (9%), including one case of uDDE and two cases of hDDE. Nonlinear DDE detection is likely hindered by the use of few density levels and small density ranges. Based on our models, DIE and +DDE promoted stable and persistent populations. uDDE and −DDE generated an Allee effect that decreases minimum patch size. Last, −DDE and hDDE models yielded bistability that allows the establishment of populations at lower densities. We conclude that the emigration process can be a diverse function of density in nature and that alternative DDE forms can have important consequences for population dynamics.
The relationship between conspecific density and the probability of emigrating from a patch can play an essential role in determining the population-dynamic consequences of an Allee effect. In this paper, we model a population that inside a patch is diffusing and growing according to a weak Allee effect per-capita growth rate, but the emigration probability is dependent on conspecific density. The habitat patch is one-dimensional and is surrounded by a tuneable hostile matrix. We consider five different forms of density dependent emigration (DDE) that have been noted in previous empirical studies. Our models predict that at the patch-level, DDE forms that have a positive slope will counteract Allee effects, whereas, DDE forms with a negative slope will enhance them. Also, DDE can have profound effects on the dynamics of a population, including producing very complicated population dynamics with multiple steady states whose density profile can be either symmetric or asymmetric about the center of the patch. Our results are obtained mathematically through the method of subsuper solutions, time map analysis, and numerical computations using Wolfram Mathematica.
We study positive solutions to a steady state reaction diffusion equation arising in population dynamics, namely, \begin{equation*} \label{abs} \left\lbrace \begin{matrix}-\Delta u=\lambda u(1-u) ;~x\in\Omega\\ \frac{\partial u}{\partial \eta}+\gamma\sqrt{\lambda}[(A-u)^2+\epsilon]u=0; ~x\in\partial \Omega \end{matrix} \right. \end{equation*} where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\); \(N > 1\) with smooth boundary \(\partial \Omega\) or \(\Omega=(0,1)\), \(\frac{\partial u}{\partial \eta}\) is the outward normal derivative of \(u\) on \(\partial \Omega\), \(\lambda\) is a domain scaling parameter, \(\gamma\) is a measure of the exterior matrix (\(\Omega^c\)) hostility, and \(A\in (0,1)\) and \(\epsilon > 0\) are constants. The boundary condition here represents a case when the dispersal at the boundary is U-shaped. In particular, the dispersal is decreasing for \(u < A\) and increasing for \(u > A\). We will establish non-existence, existence, multiplicity and uniqueness results. In particular, we will discuss the occurrence of an Allee effect for certain range of \(\lambda\). When \(\Omega=(0,1)\) we will provide more detailed bifurcation diagrams for positive solutions and their evolution as the hostility parameter \(\gamma\) varies. Our results indicate that when \(\gamma\) is large there is no Allee effect for any \(\lambda\). We employ a method of sub-supersolutions to obtain existence and multiplicity results when \(N > 1\), and the quadrature method to study the case \(N=1\).
Fragmentation creates landscape-level spatial heterogeneity which in turn influences population dynamics of the resident species. This often leads to declines in abundance of the species due to increased susceptibility to edge effects between the remnant habitat patches and the lower quality “matrix” surrounding these focal patches. In this paper, we formalize a framework to facilitate the connection between small-scale movement and patch-level predictions of persistence through a mechanistic model based on reaction–diffusion equations. The model is capable of incorporating essential information about edge-mediated effects such as patch preference, movement behavior, and matrix-induced mortality. We mathematically analyze the model’s predictions of persistence with a general logistic-type growth term and explore their sensitivity to demographic attributes in both the patch and matrix, as well as patch size and geometry. Also, we provide bounds on demographic attributes and patch size in order for the model to predict persistence of a species in a given patch based on assumptions on the patch/matrix interface. Finally, we illustrate the utility of this framework with a well-studied planthopper species (Prokelisia crocea) living in a highly fragmented landscape. Using experimentally derived data from various sources to parameterize the model, we show that, qualitatively, the model results are in accord with experimental predictions regarding minimum patch size of P. crocea. Through application of a sensitivity analysis to the model, we also suggest a ranking of the most important model parameters based on which parameter will cause the largest output variance.
We study positive solutions to the steady state reaction diffusion equation: \begin{equation*} \begin{cases} - \Delta v = \lambda v(1-v), & x \in \Omega_0, \\ \frac{\partial v}{\partial \eta} + \gamma \sqrt{\lambda} ( v-A)^2 v =0 , & x \in \partial \Omega_0, \end{cases} \end{equation*} where $\Omega_0$ is a bounded domain in $\mathbb{R}^n$; $n \ge 1$ with smooth boundary $\partial \Omega_0$, ${\partial }/{\partial \eta}$ is the outward normal derivative, $A \in (0,1)$ is a constant, and $\lambda$, $\gamma$ are positive parameters. Such models arise in the study of population dynamics when the population exhibits a U-shaped density dependent dispersal on the boundary of the habitat. We establish existence, multiplicity, and uniqueness results for certain ranges of the parameters $\lambda$ and $\gamma$. We obtain our existence and mulitplicity results via the method of sub-super solutions.
We analyze the positive solutions to \begin{equation*} \left\{ \begin{array}{cl} - \Delta v = \lambda v(1-v); & \Omega_0, \\ \frac{\partial v}{\partial \eta} + \gamma \sqrt{\lambda} v =0 ; & \partial \Omega_0, \end{array} \right. \end{equation*} where $\Omega_0=(0,1)$ or is a bounded domain in $\mathbb{R}^n$; $n =2,3$ with smooth boundary and $|\Omega_0|=1$, and $\lambda, \gamma$ are positive parameters. Such steady state equations arise in population dynamics encapsulating assumptions regarding the patch/matrix interfaces such as patch preference and movement behavior. In this paper, we will discuss the exact bifurcation diagram and stability properties for such a steady state model.
We discuss a quadrature method for generating bifurcation curves of positive solutions to some autonomous boundary value problems with nonlinear boundary conditions. We consider various nonlinearities, including positone and semipositone problems in both singular and nonsingular cases. After analyzing the method in these cases, we provide an algorithm for the numerical generation of bifurcation curves and show its application to selected problems.
We investigate the stability properties of positive steady-state solutions of semilinear initial–boundary-value problems with nonlinear boundary conditions. In particular, we employ a principle of linearized stability for this class of problems to prove sufficient conditions for the stability and instability of such solutions. These results shed some light on the combined effects of the reaction term and the boundary nonlinearity on stability properties. We also discuss various examples satisfying our hypotheses for stability results in dimension 1. In particular, we provide complete bifurcation curves for positive solutions for these examples.
